Linear Algebra

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Contents


Theorems

solution Find the eigenvalues of the matrix \begin{bmatrix}
5 & 2 \\
3 & 6 \\
\end{bmatrix}
solution Define the adjoint of a matrix.
solution Define a self-adjoint matrix.
solution Define a unitary matrix.
solution Show that (A^*)^* = A\,
solution Show that (AB^*)^* = BA^*\,
solution Show that AA^*\, is self-adjoint.
solution Show that the identity matrix I\, is self-adjoint.
solution Show that the zero matrix \mathcal{O}\, is self-adjoint.
solution Show that (\alpha A + \beta B)^* = (\overline{\alpha}A^* + \overline{\beta}B^*)\,
solution Let A\, be an n\times n matrix such that A^2=A\,, and let I\, be the n\times n identity matrix. Prove that {\rm rank}(A)+{\rm rank}(A-I)=n\,.
solution Let A\, be an m\times n matrix. Prove that {\rm Null}(A)^\perp={\rm Col}(A^T).

Matrices

Basic Problems

solution If  A=\begin{bmatrix} 0 & 1 & 2 \\ 2 & 3 & 4 \end{bmatrix}\, and  B=\begin{bmatrix} 1 & 0 & 0 \\ 2 & -3 & 1 \end{bmatrix}\, evaluate 2A+3B\,

solution Find x\, such that \begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1\\ 2 \\ x \end{bmatrix}=0\,.

solution If A=\begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}\, Show that  A^2-4A+7I=O\,

Solution If w is cube root of unity,show that {\begin{bmatrix} 1 & w & w^2 \\ w & w^2 & 1 \\ w^2 & 1 & w \end{bmatrix}+\begin{bmatrix} w  & w^2 & 1 \\ w^2 & 1 & w \\ w & w^2 & 1 \end{bmatrix}}\begin{bmatrix} 1 \\ w \\ w^2 \end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\,

solution If A=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\, for all integral values of n,show that A^n=\begin{bmatrix} \cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{bmatrix}\,

solution Find the value of determinant of the matrix \begin{bmatrix} 29 & 26 & 22 \\ 25 & 31 & 27 \\ 63 & 54 & 46 \end{bmatrix}\,

solution Show that \begin{vmatrix} bc & b+c & 1 \\ ca & c+a & 1 \\ ab & a+b & 1 \end{vmatrix}=(a-b)(b-c)(c-a)\,

solution Prove that the determinant of the matrix  \begin{bmatrix} y+z & z & y \\ z & z+x & x \\ y & x & x+y \end{bmatrix}=4xyz\,

solution Show that \begin{vmatrix} a+b & b+c & c+a \\ b+c & c+a & a+b \\ c+a & a+b & b+c \end{vmatrix}=2 \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}\,

solution Without expanding the determinant of the matrix A=\begin{bmatrix} 0 & p-q & p-r \\ q-p & 0 & q-r \\ r-p & r-q & 0  \end{bmatrix}\, prove that |A|=0\,

solution Prove that \begin{vmatrix} -2a & a+b & c+a \\ b+a & -2b & b+c \\ c+a & c+b & -2c \end{vmatrix}=4(a+b)(b+c)(c+a)\,

solution If a,b,c are distinct, abc\not\equiv 0 \, and \begin{vmatrix} a & a^3 & a^4-1 \\ b & b^3 & b^4-1 \\ c & c^3 & c^4-1 \end{vmatrix}=0\, then prove that abc(ab+bc+ca)=a+b+c\,

solution Show that \begin{vmatrix} a & a+b & a+b+c \\ 2a 3a+2b & 4a+3b+2c \\ 3a & 6a+3b & 10a+6b+3c \end{vmatrix}=a^3\,

solution Prove that \begin{vmatrix} a^2 & bc & ac+c^2 \\a^2+ab & b^2 & ac \\ab & b^2+bc & c^2 \end{vmatrix}=4a^2 b^2 c^2\,

solution Without expanding the determinant prove that \begin{vmatrix} 1 & bc & b+c \\ a & ca & c+a \\ 1 & ab & a+b \end{vmatrix}=\begin{vmatrix} 1 & a & a^2 \\1 & b & b^2 \\ 1 & c & c^2 \end{vmatrix}\,

solution Solve for x given \begin{vmatrix} x-2 & 2x-3 & 3x-4 \\ x-4 & 2x-9 & 3x-16 \\ x-8 & 2x-27 & 3x-64 \end{vmatrix}=0\,

solution Show that the determinant of the matrix \begin{bmatrix} \cos (\theta+\alpha) & \sin (\theta+\alpha) & 1 \\ \cos (\theta+\beta) & \sin(\theta+\beta) & 1 \\ \cos(\theta+\alpha) & \sin(\theta+\alpha) & 1 \end{bmatrix}\, is independent of theta.

solution Show that \begin{vmatrix} b+c & a-b & a \\ c+a & b-c & b \\ a+b & c-a & c \end{vmatrix}=3abc-a^3-b^3-c^3\,


Inverse & Rank of a Matrix

solution If A and B are two non-singular matrices of the same type,then adjoint(AB)=(adjoint B)(adjoint A)

solution If A,B are invertible matrices of the same order,then (AB)^{-1}=B^{-1} A^{-1}\,

Solution Compute the adjoint of the matrix A=\begin{bmatrix} 1 & 2 & 2 \\ 2 & 3 & 0 \\ 0 & 1 & 2 \end{bmatrix}\,

solution Find the adjoint and inverse of A=\begin{bmatrix} 2 & 3 & 4 \\ 4 & 3 & 1 \\ 1 & 2 & 4 \end{bmatrix}\,

solution Determine the rank of A=\begin{bmatrix} 4 & 2 & 3 \\ 8 & 4 & 6 \\ -2 & -1 & -1.5 \end{bmatrix}\,

solutionFind the rank of A,rank of B A=\begin{bmatrix} 1 & 5 & 4 \\ 0 & 3 & 2 \\ 2 & 3 & 10 \end{bmatrix}\,, B=\begin{bmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \end{bmatrix}\,

solution Determine the values of b such that the rank of the matrix A is 3. A=\begin{bmatrix} 1 & 1 & -1 & 0 \\ 4 & 4 & -3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3 \end{bmatrix}\,

solution Find the non-singular matrices P and Q such that the normal form of A is PAQ where A=\begin{bmatrix} 1 & 3 & 6 & -1 \\ 1 & 4 & 5 & 1 \\ 1 & 5 & 4 & 3 \end{bmatrix}\,. Hence find its rank.

solution Find P and Q such that the normal form of A=\begin{bmatrix} 1 & -1 & -1 \\ 1 & 1 & 1 \\ 3 & 1 & 1 \end{bmatrix}\, is PAQ. Hence the find the rank.

solution A=\begin{bmatrix}  1 & 5 & 4 \\ 0 & 3 & 2 \\ 2 & 3 & 10\end{bmatrix}\,, B=\begin{bmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3\end{bmatrix}\,. Find the rank of A+B\, and AB\,.

solution Solve by Cramer's rule x+y+z=11,2x-6y-z=0,3x+4y+2z=0\,

Inner Products

solution Define an inner product.
solution Show that \langle x, \alpha y\rangle = \overline{\alpha}\langle x, y\rangle
solution Show that \langle x, y + z\rangle = \langle x, y\rangle + \langle x, z\rangle
solution Show that \langle x, y\rangle + \langle y, x\rangle = 2 \mbox{ Re }\langle x, y\rangle
solution Show that \langle x, y\rangle - \langle y, x\rangle = 2i \mbox{ Im }\langle x, y\rangle
solution Show that \langle \alpha x, \beta y\rangle = \alpha\overline{\beta}\langle x, y\rangle
solution Show that \langle \alpha x, \alpha y\rangle = |\alpha|^2\langle x, y\rangle
solution Show that \langle -x, -y\rangle = \langle x, y\rangle
solution Show that \langle x, \vec{0}\rangle = 0
solution Show that \langle \vec{0}, y\rangle = 0
solution Show that \langle x, x\rangle is always real.

Vector Algebra

Vector Addition

solution If AB,BE,CF\, are the medians of a triangle,then prove that \quad \bar{AB}+\bar{BE}+\bar{CF}=\bar{O}\,

solution If G is the centroid of the triangle ABC,prove that \bar{GA}+\bar{GB}+\bar{GC}=\bar{O}\, where A,B,C\, are the vertices of the triangle ABC and \bar{O}\, is the point vector

solution The position vectors of A and B are 2\bar{i}+\bar{j}-\bar{k},\bar{i}+2\bar{j}+3\bar{k}\, respectively.Find the position vector of the point which divides the line segment AB in the ration 2:3.

solution If \bar{a}=2\bar{i}+\bar{k},\bar{b}=3\bar{i}+4\bar{k},\bar{c}=8\bar{i}+9\bar{k}\, then express \bar{c}\, as a linear combination of \bar{a}\, and \bar{b}\,.

solution If \bar{OA}=\bar{i}+\bar{j}+\bar{k},\bar{AB}=3\bar{i}+2\bar{j}+\bar{k},\bar{BC}=\bar{i}+2\bar{j}-2\bar{k},\bar{CD}=2\bar{i}+\bar{j}+3\bar{k}\,,then find the position vector of D.

solution If \bar{a}\, is the position vector whose point is (3,-2)\,.Find the coordinates of a point B such that \bar{AB}=\bar{a}\,,the coordinates of A are (-1,5)\,

solution Find a vector of magnitude 6units which is parallel to the vector \bar{i}+\sqrt{3}\bar{j}\,

solution Find the magnitude of the vector 7\bar{i}-3\bar{j}+5\bar{k}\,

solution If the position vectors of A and B are 2\bar{i}-9\bar{j}-4\bar{k},6\bar{i}-3\bar{j}+8\bar{k}\, respectively,find the unit vector in the direction of AB.

solution If the position vectors of A and B are \bar{i}+3\bar{j}-7\bar{k},5\bar{i}-2\bar{j}+4\bar{k}\, respectively,determine the direction cosines of \bar{AB}\,

solution In a triangle ABC if A=2\bar{i}+4\bar{j}-\bar{k},B=4\bar{i}+5\bar{j}+\bar{k},C=3\bar{i}+6\bar{j}-3\bar{k}\, and D is the mid point of the side BC, then find the length of AD.

solution Show that the points represented by \bar{i}+2\bar{j}+3\bar{k},3\bar{i}+4\bar{j}+7\bar{k},-3\bar{i}-2\bar{j}-5\bar{k}\, are collinear.

solution Show that the points A,B,C,D with position vectors 6\bar{i}-7\bar{j},16\bar{i}-19\bar{j}-4\bar{k},3\bar{j}-6\bar{k},2\bar{i}+5\bar{j}+10\bar{k}\, are not coplanar.

solution Prove that three points whose vectors are \bar{i}+2\bar{j}+3\bar{k},-\bar{i}-\bar{j}+8\bar{k},4\bar{i}+4\bar{j}+6\bar{k}\, form an equilateral triangle.

solution Show that the triangle ABC whose vertices are 7\bar{i}+10\bar{k},-\bar{i}+6\bar{j}+6\bar{k},-4\bar{i}+9\bar{j}+6\bar{k}\, is isoscles and right angled.

solution Obtain the point of intersection of the line joining the points \bar{i}-2\bar{j}-\bar{k},2\bar{i}+3\bar{j}+\bar{k}\, with the plane through the points 2\bar{i}+\bar{j}-3\bar{k},4\bar{i}-\bar{j}+2\bar{k}\, and 3\bar{i}+\bar{k}\,

Vector Product

solution Show that the points whose position vectors are 2\bar{i}-\bar{j}+\bar{k},\bar{i}-3\bar{j}-5\bar{k},3\bar{i}-4\bar{j}-4\bar{k}\, are the vertices of a right angled triangle.

solution Find a vector \bar{d}\, which is perpendicular to both \bar{a}=4\bar{i}+5\bar{j}-\bar{k},\bar{b}=\bar{i}-4\bar{j}+5\bar{k}\, and \bar{d}\cdot \bar{c}=21\, where\bar{c}=3\bar{i}+\bar{j}-\bar{k}\,

solution If \bar{a},\bar{b},\bar{c}\, are mutually perpendicular vectors of equal magnitude,show that \bar{a}+\bar{b}+\bar{c}\, is equally inclined to \bar{a},\bar{b},\bar{c}\,

solution Dot products of the vectors \bar{i}+\bar{j}-3\bar{k},\bar{i}+3\bar{j}-2\bar{k},2\bar{i}+\bar{j}+4\bar{k}\, are 0,5,8\, respectively.Find the vector.

solution Find the vector equation of a plane which is at a distance of 5units from the origin and which has 2\bar{i}+3\bar{j}+6\bar{k}\, as a normal vector.

solution Find the vector equation of the plane through the point A(3,-2,1)\, and perpendicular to the vector 4\bar{i}+7\bar{j}-4\bar{k}\,.

solution Find the equation of the plane passing through the point (3,4,5)\, and parallel to the plane \bar{r}\cdot(2\bar{i}+3\bar{j}-\bar{k})=6\,

solution If \bar{a}=2\bar{i}-3\bar{j}+5\bar{k},\bar{b}=-\bar{i}-4\bar{j}+2\bar{k}\,,then write \bar{a}\times\bar{b}\,

solution Determine the unit vector perpendicular to both the vectors 2\bar{i}+\bar{j}+3\bar{k},\bar{i}-2\bar{j}+\bar{k}\,

solution Find the vector area of a parallelogram whose diagonals are determined by the vectors \bar{a}=3\bar{i}+\bar{j}-2\bar{k},\bar{b}=\bar{i}-3\bar{j}+4\bar{k}\,

solution Find a vector of magnitude 3 and which is perpendicular to both the vectors 3\bar{i}+\bar{j}-4\bar{k},6\bar{i}+5\bar{j}-2\bar{k}\,

solution IF (1,2,3),(2,5,-1),(-1,1,2)\, are the vertices of a triangle, find its area.

solution Find a unit vector perpendiculars to the plane ABC where A=(3,-1,2),B(1,-1,-3),C(4,-3,1)\,

solution Let \bar{a}=2\bar{i}+3\bar{j}+4\bar{k},\bar{b}=\bar{i}-2\bar{j}+\bar{k}\,.If a vector \bar{r}\, satisfies \bar{a}\times\bar{r}=3\bar{b}\, and \bar{a}\cdot\bar{r}=2\, then find the vector \bar{r}\,

solution Find the volume of the parallelopiped whose edges are \bar{i}+\bar{j}+\bar{k},\bar{i}-\bar{j}+\bar{k},\bar{i}+\bar{j}-\bar{k}\,.

solution Show that the four points having position vectors 6\bar{i}-7\bar{j},16\bar{i}-19\bar{j}-4\bar{k},3\bar{i}-6\bar{k},2\bar{i}+5\bar{j}+10\bar{k}\, are not coplanar.

solution If the two vectors \bar{a}=2\bar{i}+\bar{j}+2\bar{k},\bar{b}=5\bar{i}-3\bar{j}+\bar{k}\, are two vectors,find the projection of \bar{b}\, on \bar{a}\,

solution Reduce the equation \bar{r}\cdot(4\bar{i}-12\bar{j}-3\bar{k})+7=0\, to normal form and hence find the length of the perpendicular from the origin to the plane.

solution Find the angle between the planes \bar{r}\cdot(2\bar{i}-\bar{j}+\bar{k})=6,\bar{r}\cdot(\bar{i}+\bar{j}+2\bar{k})=7\,

solution Find the value of lambda for which the four points with position vectors 3\bar{i}-2\bar{j}-\bar{k},2\bar{i}+3\bar{j}-4\bar{k},-\bar{i}+\bar{j}+2\bar{k},4\bar{i}+5\bar{j}+\lambda\bar{k}\, are coplanar.

solution Find the volume of the tetrahedron with vertices (1,1,3),(4,3,2),(5,2,7),(6,4,8)\,

solution Find the vector equation of the line passing through three non-collinear points -2\bar{i}+6\bar{j}-6\bar{k},-3\bar{i}+10\bar{j}-9\bar{k},-5\bar{i}-6\bar{k}\,.Also find its cartesian equation.

solution Find the equation of the plane passing through the points 3\bar{i}-5\bar{j}-\bar{k},-\bar{i}+5\bar{j}+7\bar{k}\, and parallel to 3\bar{i}-\bar{j}+7\bar{k}\,

solution If \begin{vmatrix} a & a^2 & 1+a^3 \\ b & b^2 & 1+b^3 \\ c & c^2 & 1+c^3 \end{vmatrix}=0\, and the vectors \bar{A}=(1,a,a^2),\bar{B}=(1,b,b^2),\bar{C}=(1,c,c^2)\, are non-coplanar,then prove that abc=-1\,

solution Find the perpendicular distance from the origin to the plane passing through the points (1,-2,5),(0,-5,-1),(-3,5,0)\,

Jordan

solution \begin{bmatrix} 2 & 0 & -1 & 1 \\ 1 & 1 & -1 & 1 \\ 1 & -3 & 2 & -1 \\ 0 & -3 & 2 & -1 \end{bmatrix}


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